Low-lying zeros of quadratic Dirichlet $L$-functions: Lower order terms for extended support

TL;DR

This paper analyzes the distribution of low-lying zeros of quadratic Dirichlet L-functions, providing an asymptotic expansion with lower order terms and identifying a phase transition at support boundary.

Contribution

It offers a new asymptotic expansion for the 1-level density of zeros, revealing a phase transition and a novel lower order term at support boundary.

Findings

Asymptotic expansion valid for support in (-2,2)

Identification of phase transition at support boundary 1

Discovery of a new lower order term involving hat phi(1)

Abstract

We study the $1$-level density of low-lying zeros of Dirichlet $L$-functions attached to real primitive characters of conductor at most $X$. Under the Generalized Riemann Hypothesis, we give an asymptotic expansion of this quantity in descending powers of $\log X$, which is valid when the support of the Fourier transform of the corresponding even test function $\phi$ is contained in $(-2,2)$. We uncover a phase transition when the supremum $\sigma$ of the support of $\hat \phi$ reaches $1$, both in the main term and in the lower order terms. A new lower order term appearing at $\sigma=1$ involves the quantity $\hat \phi (1)$, and is analogous to a lower order term which was isolated by Rudnick in the function field case.